Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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31 views

A variation of Cesaro means

It follows from the Cesaro means that $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n g(k)=c~~~{\rm if}~~~\lim_{k\rightarrow\infty}g(k)=c.$$ Instead, let us consider a sequence $\{f_n\}$ such that ...
0
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4answers
39 views

$E_n - A$ invertible if $\|A \|< 1$

Let $E_n$ be the identity matrix and $\|\cdot \|$ a matrix norm. How to prove with the help of Banach's fixed-point theorem that $E_n - A$ is invertible if $\|\,A\,\| < 1$?
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2answers
25 views

Diffeomorphism of elementary symmetric polynomials

Let $\sigma_1(x,y,z) = x + y + z$, $\sigma_2(x,y,z) = xy + xz + yz$ and $\sigma_3(x,y,z) = xyz$. When is the map $\Phi: \mathbb{R}^3 \to \mathbb{R}^3, \Phi\,(x,y,z) = \begin{pmatrix} \sigma_1(x,y,z) \...
4
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1answer
54 views

Integral inequality with gradient

Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
4
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2answers
136 views

Showing the set of real values for which the pre-image has measure greater than zero is measure zero

The question is stated as follows: Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is measurable, then the set $E = \{x \in \mathbb{R} \ | \ m(f^{-1}(x)) > 0 \}$ has measure zero. This ...
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0answers
46 views

contraction mapping proof

I was reading a paper, and there is a proof I don't understand. How did they get from eq(6.4) to eq(6.5) using the norm they defined? Any help would be appreciated. Thanks in advance! Here is the ...
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0answers
31 views

Limit of weighed partial sums

Let $f_n(k)>0$ be a non-negative function, defined on $k\in\{0,1,2,\dots,n\}$, such that $\lim_{n\rightarrow\infty}f_n(k)=f(k)$ and $\lim_{k\rightarrow\infty}f(k)=0$. Also, let $w_n(k)$ be a ...
0
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1answer
39 views

Existence and Smoothness of Vector Calculus Identities [closed]

How to proof there exist smooth and globally defined solutions to all Vector calculus indentities ? For example: proof there exist smooth and globally defined solutions to the divergence of the curl ...
0
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4answers
118 views

Proving that ${k+x \choose 2k + 1}=-{k-x \choose 2k + 1} $

I need to prove the following, someone can help me? ${k+x \choose 2k + 1}=-{k-x \choose 2k + 1} $ I tried the following: $\frac{(k+x)!}{(2k + 1)!((k+x)-(2k+1))!} = -1\frac{(k-x)!}{(2k + 1)!((k-x)-(...
2
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2answers
46 views

Existence of a function that fulfills an equation

I am just revising for my exams and came across this question: Show that in the Banach-space of functions that are continuous in the interval $[-1,1]$, together with the supremum-norm, there is ...
1
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1answer
48 views

Uniform continuity supremum

In the User's Guide to Viscosity Solutions, it is claimed at the top of page 31 that If $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ is a uniformly continuous function, there exists a positive ...
5
votes
3answers
248 views

Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
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2answers
61 views

Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
0
votes
1answer
47 views

Math needed to study Navier-Stokes existence and smoothness problem [closed]

This is Navier-Stokes existence and smoothness problem. I think the main problem is that I am not familiar with the mathematics of the Navier-Stokes existence smoothness problem. What kind of math ...
7
votes
1answer
107 views

$\|D_{f}(x) v\|=\|v\|$ $\implies$ $f$ is an isometry

Let $f\colon \mathbb{R}^m \to \mathbb{R}^m$ be a $C^2$ map such that $\|D_f(x)v\|=\|v\|$ for all $v\in\mathbb{R}^m$, where $D_f(x)$ is the derivative of $f$ at $x$. Then I am asked to prove that $\|f(...
1
vote
1answer
42 views

Is $\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}$ bounded independently of $a$?

Fix an $a\in \mathbb{R}$ and consider the sum $$\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}.$$ Is this sum bounded independent of $a$? I think the answer should be yes since for $...
0
votes
1answer
36 views

Suppose a sequence's subsequences have at least one subsubsequence that converges almost surely to $X$. Prove convergence in probability

Probability with Martingales What I tried: 'only if' Suppose a sequence converges in probability to $X$. By $d$ there exists a subsequence that converges almost surelyto $X$. Then by $a$, ...
4
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1answer
164 views

Existence of a homeomorphism that does not return much

Let $f:X\rightarrow X$ a homeomorphism where $X$ is a compact metric space. Fix $x\in X$, denote $O(f,x)=\{ f^n(x):n\in \mathbb{Z}\}$ the orbit of $f$ by $x$. For $m\in \mathbb{N}$ denote $O(f,x,m)...
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0answers
28 views

Estimate the Sobolev norm of negative order of a function

I would like to estimate the Sobolev norm of order $-1$ of the function $f(x)$ which is defined as follows: Let $\psi\in C^\infty_c(\mathbb{R})$ be a compactly supported smooth function on the real ...
0
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1answer
21 views

This is a question about the best type of regression analysis to use in my software

Let me start by saying that I am not a mathematician and I am not very good at math. I am mainly interested in obtaining the best possible results. I am currently doing trial and error with my ...
2
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1answer
113 views

What branch/field of mathematics is this? [closed]

I do not want solutions, I just want the field/branch of mathematics that these problems deal with, and possibly a good online source or two to learn it. Problems :- 1:- 2:- 3:- 4:- ...
2
votes
1answer
39 views

A continuous onto/surjective function from $[0, 1) \to \Bbb R$.

Does there exist a continuous onto/surjective function from $[0, 1) \to \Bbb R$? Finding difficult to site an example...
1
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2answers
72 views

What is the diagonal principle?

I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of $(c)$). What is the diagonal principle? Is that related to Cantor's ...
3
votes
4answers
54 views

How to show that $x \sin \frac{\pi}{x} > \pi \cos \frac{\pi}{x}$ for $x \in (1, \infty)$?

How could I prove that $x \sin \frac{\pi}{x} > \pi \cos \frac{\pi}{x}$ for $x \in (1, \infty)$? Dividing both sides through by $x \sin \frac{\pi}{x}$ and letting $y = \frac{\pi}{x}$ gives the ...
1
vote
1answer
28 views

Maximum variance

Consider a random variable $X$ with continuous probability density $f(x)$ and compact support, say $[a,b]$ with $a<b$. Moreover, let $f(x)$ vanish at the boundary, i.e. $f(a) = f(b) = 0$. ...
1
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1answer
35 views

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$. Show that $F(\cdot , t_0)$ has exactly one zero using the Implicit Function Theorem

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$ is a through $t$ parametrized family of polynominals. $a_i : I \to \Bbb R \:\:\:\mathrm{ are }\: \mathcal C^k$- functions with $k \ge 1$. Let $x_0$ be a zero ...
2
votes
2answers
42 views

$n\lvert a_n\rvert \to 0$, then $\sum_{k=1}^n \frac k n\lvert a_k\rvert \to 0$?

I am now working on the converse of Abel's THM and found out one proof of the conditional converse of the theorem. The proof says, Suppose $$n\lvert a_n\rvert \to 0$$ as $$n \to \infty$$ Then, $$\...
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0answers
34 views

Computing $\delta$ as a function of $\epsilon$ for uniformly continuous functions

So a function is uniformly continuous if $\forall \epsilon > 0, \exists \delta >0 $ s.t. $\forall x,y$, $|x-y|<\delta \rightarrow |f(x)-f(y)| < \epsilon$. Is there another category of ...
0
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1answer
10 views

Finding a 2-chain whose boundary is $c - c_{1, n}$ for a 1-cube $c$ such that $c(0) = c(1)$ and a circle $c_{1, n}$.

Problem (Spivak, 4-24). Define $c_{1, n} : [0, 1] \to \mathbb{R}^2$ by $c_{1, n}(t) = (\cos 2\pi nt, \sin 2\pi nt)$. If $c$ is a singular 1-cube in $\mathbb{R}^2-\{ 0 \}$ with $c(0) = c(1)$, show ...
2
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2answers
57 views

Is this set countable or uncountable? (Related to mean value theorem)

Let $f$ be a differentiable function on the real line. Consider any $h\in (0,1)$. By the mean value theorem there exists $d_h\in (0,h)$ such that $f'(d_h)=\frac{f(h)-f(0)}{h}$. Is the set $\{d_h\}$ ...
3
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1answer
70 views

Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...
0
votes
1answer
31 views

Smooth Approximations in $L^2((0,1))$

Let $L^2((0,1))$ be as usual the Lebesgue space of measurable complex-valued functions $f:(0,1) \rightarrow \mathbb{C}$ such that $\int |f(x)|^2 dx < \infty$. It is a well known fact (see e.g Lieb ...
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0answers
22 views

A $C^1$ function in Orlicz Sobolev space

How to prove that thi functional is $C^1$: $$ I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx $$ Where $\Phi$ is an N-function and $F(t)=\int_{0}^t f(s) ds$ where ...
2
votes
1answer
79 views

Good, relatively short math textbooks? [closed]

Recently I've been trying to decide on some fun math summer reading on some areas of math which I have less experience with. I'm an undergrad studying mathematics with a focus in actuarial science, ...
1
vote
1answer
119 views

Closure of $\frac {1} {n} $ [duplicate]

I have as a definition of the closure of a set $E$ in a metric space $(S,d)$ that $E^-$ is the intersection of all closed sets containing E (Elementary analysis the theory of calculus by Kenneth Ross)....
1
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1answer
53 views

Why is sum $\cot^{-1}(n^2+n+1)$ equal to $\cot^{-1}\left(\frac1{m+1}\right)$?

Can u Please Explain This sum result : $$\sum_{n=0}^m \cot^{-1} (n^2+n+1) = \cot^{-1} \left(\frac1{m+1}\right)$$
2
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1answer
33 views

$f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^1$ such that $||f(x) - f(y)|| \geq c||x-y||$ is a diffeomorphism.

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ map. Suppose that exists $c>0$ such that $||f(x)-f(y)||\geq c||x-y||$ for all $x,y \in \mathbb{R}^n$. Prove that $f$ is a diffeomorphism. I ...
1
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1answer
35 views

How prove that the range of an rectifiable curve has measure zero?

Let $f:[a,b] \rightarrow \mathbb{R}^n$ be a rectifiable continuous curve, show that $f[a,b]$ has content zero. If $f$ is continuous then is integrable so we have that $[a,b]\times f[a,b]$ has ...
0
votes
1answer
25 views

Integral on $\partial B(0,R)$

I want to calculate $\int_{\partial B(x,\epsilon)}\frac{1}{|x-y|^2}$ with $B(x,\epsilon)\in \mathbb{R}^3$ Time ago I saw a paper who said (if I am right) $\int_{\partial B(0,R)}\frac{1}{|x-y|^{n-1}}=...
0
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3answers
64 views

Finding $n$th derivative in an unusual way

If $f(z) = \frac{e^{iz}}{z^2-1}$, then $f^{(4)}(z)$ can be found by differentiating $f(z)$ four times. I tried to use Cauchy's integral formula, but the integrand is not holomorphic at $z=0$, so we ...
0
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2answers
29 views

$f(x)=x\ln x-\frac{k}{x}$ and $f(x_1)=f(x_2)=0$ $\Rightarrow$ $f'\left(\frac{x_1+x_2}{2}\right)\not=0$

If $f(x)=x\ln x-\frac{k}{x}$. And $x_1$, $x_2$ are two roots of $f(x)=0$. Then $f'\left(\frac{x_1+x_2}{2}\right)\not=0$ First, I determine the range of $k$. Because $f=0$ has two roots $\iff$ $x^2\...
3
votes
1answer
31 views

Differentiable, nonnegative derivative ae, nondecreasing?

I have been trying to prove or disprove: Let $g$ be differentiable on the reals, have $g'(x)\geq 0$ except countably many values, then $g$ is non-decreasing. The main problem I face is that I don't ...
0
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1answer
25 views

Finding Taylor series without using derivatives

If $\displaystyle f(z) = \frac{e^{iz}}{z^2-1}$ then we can set $g(z)=e^{iz}$ and $h(z)=z^2-1$. The Maclaurin expansion for $e^{iz}$ is $$\sum\limits_{n=0}^\infty \frac{(iz)^n}{n!}$$ so $\displaystyle ...
0
votes
1answer
44 views

Is there a function $f$ satisfying this integrability condition?

I am wondering if it is possible to find a real valued function $f$ such that $$ \int_{0}^{1} \frac{1}{x^2 T(f(x))} \ dx < \infty $$ where $T(x)=\cos(x)$ or $T(x)=\sin(x)$. Thanks.
3
votes
2answers
49 views

How did Euler prove the partial fraction expansion of the cotangent function?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I've seen several modern proofs of it and they all ...
4
votes
2answers
90 views

Nonexistence of a continuous injection $f:S^2 \rightarrow \mathbb{R^2}$

What is the "easiest" way to show that there is no continuous injection $f:S^2 \rightarrow \mathbb{R^2}$? Sure the Borsuk-Ulam theorem implies that result, but this may be a "difficult" way.
0
votes
0answers
25 views

Product of p-summable sequences

Let $1<p<\infty$ and $(a_n)_{n=0}^\infty$ and $(b_n)_{n=0}^\infty$ be two complex, $p$-summable sequences, which means that \begin{equation} \sum_{n=1}^\infty |a_n|^p < \infty, \sum_{n=1}^\...
0
votes
1answer
28 views

Theorem on arithmetic of natural numbers.

From "Analysis I"-Herbert & Joachim: (starting from the Peano axioms) "There are operations addition + , multiplication · and a partial order ≤ on N which are uniquely determined by the ...
0
votes
2answers
35 views

To show $f$ conatined in oval is constant

Let $\Omega \subset \mathbb{C}$ is connected open and let $f\in O(\Omega)$ Suppose $f(\Omega) \subset L$ where $L:=\{x+iy \in \mathbb{C} \vert x^{2k} + y^{2k} = 1 \}$ for $k \in \mathbb{Z}^{+}, k >...
-2
votes
2answers
59 views

analysis exercise: show that this series converges [closed]

this This is analysis exam problem. If $\sum_n{x_n}$ converges, then $\sum_n{x_n\over n}$ converges. I don't know how to show that.